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Portfolio Credit Risk: A model of correlated credit losses dynamics and the inverse-gamma approximation

by Ridha M. Mahfoudhi of Laval University & National Bank of Canada

August 2008

Abstract: 'Top-down' approach of portfolio credit risk lacks of transparency in terms of explicit implications at the 'down' level. 'Bottom-up' approach is less flexible to accommodate for modeling the dynamics of credit loss. In this paper, a hybrid approach is proposed, which combines both the modeling parsimony of 'top-down' models and the explicit implications of 'bottom-up' models in terms of default risk of single names. Correlated forward credit losses of the n constituents of the pool (or index) are modeled under a HJM-like framework. This allows for arbitrage-free dynamics of single names' defaults matching the n single names' CDS spread curves and an explicit characterization of a name-sensitive credit loss correlation structure. Under an explicit setup of correlated lognormal cumulated credit losses, it is argued that the Inverse-Gamma distribution (with time-varying shape and scaling parameters generated back at the 'down' level by the HJM-like model) approximates the credit pool loss distribution, which results in a simple closed-form solution of CDO spreads. Calibration is made efficiently thanks to a name grouping technique and shows a high fitting power of the model, particularly for the skew of the market CDO spreads. Calibration procedure, model extensions and numerical examples illustrating the impact of the subprime crisis are discussed. Furthermore, Monte Carlo methods for the simulation of the Inverse-Gamma pool loss dynamics are developed to price CDO derivatives with early-exercise-style and/or loss-trigger features.

JEL Classification: G13.

Keywords: Portfolio credit loss, CDO, Credit correlation.

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